Causal analysis within the framework of structural autoregressive - PowerPoint PPT Presentation

Causal analysis within the framework of structural autoregressive models Alessio Moneta Scuola Superiore Sant’Anna a.moneta@santannapisa.it Universitat Rovira i Virgili 13 February 2018

Outline 1. Why VAR models? 2. From VAR to SVAR models. 3. Problem of identification: two solutions based on causal search methods: ◮ Graphical Causal Models ◮ Independent Component Analysis

Why VAR models?

The Vector Autoregressive Model Given a vector y t of k variables: y t = µ t + D 1 y t − 1 + D 2 y t − 2 + . . . + D p y t − p + u t where D i are ( i = 1 , . . . , p ) are ( k × k ) matrices u t is a ( k × 1 ) vector of error terms (residuals), which are white noise and E ( u t u ′ t ) = Σ u . µ t is a ( k × 1 ) vector of constants (possibly including a deterministic trend). ⊲ Cfr. Sims, C. (1980) “Macroeconomics and Reality” Econometrica .

Structural equations and causality ⊲ Cowles Commission Approach: dominant approach in macro-econometrics 1940s-1960s ◮ Haavelmo’s “The Probability Approach in Econometrics” (1944) ◮ Economic theory dictates the causal structure ◮ If the structure is adequate: error terms conform to standard probabilistic properties (independence and normality) ◮ Measuring the strengths of causal linkages.

Example of structural model ⊲ Example: (Hoover 2006) m = α y + ε m (1) y = β m + ε y , (2) where m ≡ money, y ≡ GDP (both in logs). ⊲ Statistical properties of ε m and ε y tell whether the model is goods specified. ⊲ can we get α , β , ε m , ε y from the data? ⊲ Cfr. Hoover, K.D. (2006), “Economic Theory and Causal Inference”, in Handbook of the Philosophy of Economics .

Example of structural model ⊲ Problem of identification solved introducing two exogenous variables: m = α y + δ r + ε m (3) y = β m + γ p + ε y , (4) where r ≡ interest rate, p ≡ price level (both in logs). ⊲ can we get α , β , δ , γ ε m , ε y from the data? ⊲ Yes. Model identified under the assumption that p is not a direct cause of m and r is not a direct cause of y ⊲ Omiting the error terms: r − → m ← → y ← − p

Reduced form model ⊲ Afer substituting eq. (3) into eq. (4) and simplifying: αγ δ α 1 m = 1 − αβ p + 1 − αβ r + 1 − αβ ε m + 1 − αβ ε y (5) γ βδ β 1 y = 1 − αβ p + 1 − αβ r + 1 − αβ ε m + 1 − αβ ε y (6)

Reduced form model ⊲ Afer substituting eq. (3) into eq. (4) and simplifying: αγ δ α 1 m = 1 − αβ p + 1 − αβ r + 1 − αβ ε m + 1 − αβ ε y (5) γ βδ β 1 y = 1 − αβ p + 1 − αβ r + 1 − αβ ε m + 1 − αβ ε y (6) ⊲ This system of reduced-form equations can be now estimated, since on the r.h.s. there are only exogenous variables: m = a p + b r + u m (7) y = c p + d r + u y (8)

Reduced form model ⊲ Afer substituting eq. (3) into eq. (4) and simplifying: αγ δ α 1 m = 1 − αβ p + 1 − αβ r + 1 − αβ ε m + 1 − αβ ε y (5) γ βδ β 1 y = 1 − αβ p + 1 − αβ r + 1 − αβ ε m + 1 − αβ ε y (6) ⊲ This system of reduced-form equations can be now estimated, since on the r.h.s. there are only exogenous variables: m = a p + b r + u m (7) y = c p + d r + u y (8)

Reduced form model ⊲ Afer substituting eq. (3) into eq. (4) and simplifying: αγ δ α 1 m = 1 − αβ p + 1 − αβ r + 1 − αβ ε m + 1 − αβ ε y (5) γ βδ β 1 y = 1 − αβ p + 1 − αβ r + 1 − αβ ε m + 1 − αβ ε y (6) ⊲ This system of reduced-form equations can be now estimated, since on the r.h.s. there are only exogenous variables: m = a p + b r + u m (7) y = c p + d r + u y (8)

Reduced form model ⊲ Afer substituting eq. (3) into eq. (4) and simplifying: αγ δ α 1 m = 1 − αβ p + 1 − αβ r + 1 − αβ ε m + 1 − αβ ε y (5) γ βδ β 1 y = 1 − αβ p + 1 − αβ r + 1 − αβ ε m + 1 − αβ ε y (6) ⊲ This system of reduced-form equations can be now estimated, since on the r.h.s. there are only exogenous variables: m = a p + b r + u m (7) y = c p + d r + u y (8)

Reduced form model ⊲ Afer substituting eq. (3) into eq. (4) and simplifying: αγ δ α 1 m = 1 − αβ p + 1 − αβ r + 1 − αβ ε m + 1 − αβ ε y (5) γ βδ β 1 y = 1 − αβ p + 1 − αβ r + 1 − αβ ε m + 1 − αβ ε y (6) ⊲ This system of reduced-form equations can be now estimated, since on the r.h.s. there are only exogenous variables: m = a p + b r + u m (7) y = c p + d r + u y (8)

Reduced form model ⊲ Afer substituting eq. (3) into eq. (4) and simplifying: αγ δ α 1 m = 1 − αβ p + 1 − αβ r + 1 − αβ ε m + 1 − αβ ε y (5) γ βδ β 1 y = 1 − αβ p + 1 − αβ r + 1 − αβ ε m + 1 − αβ ε y (6) ⊲ This system of reduced-form equations can be now estimated, since on the r.h.s. there are only exogenous variables: m = a p + b r + u m (7) y = c p + d r + u y (8)

Reduced form model ⊲ Afer substituting eq. (3) into eq. (4) and simplifying: αγ δ α 1 m = 1 − αβ p + 1 − αβ r + 1 − αβ ε m + 1 − αβ ε y (5) γ βδ β 1 y = 1 − αβ p + 1 − αβ r + 1 − αβ ε m + 1 − αβ ε y (6) ⊲ This system of reduced-form equations can be now estimated, since on the r.h.s. there are only exogenous variables: m = a p + b r + u m (7) y = c p + d r + u y (8) a , ˆ c , ˆ ⊲ From the OLS estimates ˆ b , ˆ d , ˆ u m , ˆ u y , using the respective

Reduced form model ⊲ Afer substituting eq. (3) into eq. (4) and simplifying: αγ δ α 1 m = 1 − αβ p + 1 − αβ r + 1 − αβ ε m + 1 − αβ ε y (5) γ βδ β 1 y = 1 − αβ p + 1 − αβ r + 1 − αβ ε m + 1 − αβ ε y (6) ⊲ This system of reduced-form equations can be now estimated, since on the r.h.s. there are only exogenous variables: m = a p + b r + u m (7) y = c p + d r + u y (8) a , ˆ c , ˆ ⊲ From the OLS estimates ˆ b , ˆ d , ˆ u m , ˆ u y , using the respective

Instrumental variables ⊲ Notice that the structural-form coefficients ( α, β in the ex.) could be equivalently obtained by instrumental variables estimation. ⊲ Following the previous example: ◮ p : instrumental variable for eq. (1) ( m = α y + ε m ) ◮ r : instrumental variable for eq. (2) ( y = β m + ε y ) ⊲ Two-stage least squares estimation for eq. (1): 1 OLS regression of y on p : obtain ˆ y 2 OLS regression of m on ˆ y : obtain ˆ α IV

Summary on Structural Equation Modeling ⊲ SEM: provide quantitative assessment of cause-effect relationships. ⊲ Interpretation of causality: ◮ counterfactual ◮ manipulability ⊲ Probabilistic methods are used to measure causality and partially to test it, but not for the sake of causal discovery. ⊲ Dependence on a priori economic theory: ◮ necessity of identifying restrictions. ⊲ Empiricist query: where does the economic theory come from?

The crisis of the Cowles Commission approach ⊲ Up to the 1970s: consensus on the Cowles Commission approach ◮ Important economic events in the 1970s: Oil crisis (1973), stagflation. Increasing skepticism towards so-called “Keynesian Macroeconomics”. ⊲ Two major critiques: ◮ Lucas Critique (1976) on economic policy evaluation structural equations are ineffective for policy evaluation: they are unstable under intervention ◮ Sims (1980) article Macroeconomics and Reality restrictions used in the Cowles Commission approach are incredible, i.e. empirically not validated ⊲ Cfr. Favero, C.A. (2001) Applied Macroeconometrics , OUP.

The crisis of the Cowles Commission approach Reactions to the criticisms: ⊲ Change the theory maintaining the structural, theory-driven approach ◮ cfr. New Classical Macroeconomics (Lucas, Sargent), and rational expectations models ◮ even more extreme theory-driven approach: calibration ⊲ Adopt a more data-driven approach: Time-series econometric models, more intensive use of statistical methods ◮ Granger Causality (1969) ◮ Vector Autoregressive Models (VAR) (Sims 1980)

Granger causality Consider two time series X t and Y t . The idea of Granger’s (1969, 1980) causality: ◮ { Y t } causes { X t } if Y t helps to predict X t + h . Definition (G.-causality, Granger 1980: 330 ): ◮ Y t is said to cause X t + 1 if P ( X t + 1 ∈ A | Ω t ) � = P ( X t + 1 ∈ A | Ω t \ Y t ) , for a set A and for a set Ω t of “relevant information” available at time t ( Ω t \ Y t := relevant information except Y t ). Merely probabilistic notion of causality. Importance to appraise usefulness of G-causality but differences with structural causality.

T esting G-non-causality Granger non-causality (GNC) tests: ◮ Regression of X t + 1 on Ω t (including X t , Y t and their lagged values) + same regression excluding Y t (and its lagged values). ◮ Check error variances of the two regressions (with and without Y t ); and/or ◮ Test whether the Y t coefficients in the regression of X t + 1 on Ω t are zero ( F or t test).

Structural vs. G-causality (Example from Hoover 2001: 151) Suppose the unobserved structural model (DGP) is: Y t = θ X t + β 11 Y t − 1 + β 12 X t − 1 + ǫ 1 t (9) X t = γ Y t + β 21 Y t − 1 + β 22 X t − 1 + ǫ 2 t (10) Reduced form model: Y t = a 11 Y t − 1 + a 12 X t − 1 + ν 1 t (11) X t = a 21 Y t − 1 + a 22 X t − 1 + ν 2 t , (12) (coefficients in (11)-(12) are functions of coefficients in (9)-(10)) GNC tests: ◮ if a 12 = 0, then X does not G-causes Y ◮ if a 21 = 0, then Y does not G-causes X .